Kadomtsev-Petviashvili type equation for uneven water depths

A Kadomtsev-Petviashvili type equation with improved dispersion characteristics is derived for varying water depths. Linear shoaling characteristics of the equation for unidirectional waves are in accord with the principle of energy flux concept. A finite-difference scheme is devised to demonstrate the extended range of applications of the newly derived equation for a number of well-known cases. In addition, a permanent wave form with singularity is shown to be an exact solution of the KdV or unidirectional KP equations. The same form for an arbitrary direction satisfies the combined Boussinesq equations of any type.